A Separation in Modulus Property of the Zeros of a Partial Theta Function

We consider the partial theta function θ ( q , z ) : = ∑ j = 0 ∞ q j ( j + 1 ) / 2 z j , where z ∈ ℂ is a variable and q ∈ ℂ, 0 < | q | < 1, is a parameter. Set α 0 : = 3 / 2 π = 0.2756644477... We show that, for n ≥ 5, for | q | ≤ 1 − 1/(α 0 n ) and for k ≥ n there exists a unique zero ξ k of θ ( q ,.) satisfying the inequalities | q | − k +1/2 < |ξ k | < | q | −k−1/2 ; all these zeros are simple ones. The moduli of the remaining n −1 zeros are ≤ | q | −n+1/2 . A spectral value of q is a value for which θ ( q ,.) has a multiple zero. We prove the existence of the spectral values 0.4353184958... ± i 0.1230440086... for which θ has double zeros −5.963... ± i 6.104...